SOLUTION: Find the range with algebra (no graphing) of f(x)/g(x) if f(x) = sqrt{4 - x^2} g(x) = sqrt{9 - x^2}

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Question 1208157: Find the range with algebra (no graphing) of f(x)/g(x) if
f(x) = sqrt{4 - x^2}
g(x) = sqrt{9 - x^2}
Found 4 solutions by Edwin McCravy, mccravyedwin, greenestamps, math_tutor2020:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
%22f%28x%29%22%2F%22g%28x%29%22%22%22=%22%22sqrt%284-x%5E2%29%2Fsqrt%289-x%5E2%29%22%22=%22%22sqrt%28%284-x%5E2%29%2F%289-x%5E2%29%29%22%22=%22%22

The minimum value is when the numerator is 0, or when x=%22%22+%2B-+2, which
is when %22f%28x%29%22%2F%22g%28x%29%22 = 0


                        4/9 
9 + 0x - x2) 4 + 0x -   9x2 
             4 + 0x - 4/9x2
                    -77/9x2


sqrt%284%2F9+-+expr%2877%2F9%29x%5E2%2F%289-x%5E2%29%29

Thus the maximum value of %22f%28x%29%22%2F%22g%28x%29%22 is when we subtract the least
amount possible from the 4/9 under the radical, which is when x=0:

 sqrt%284%2F9+-+expr%2877%2F9%290%5E2%2F%289-0%5E2%29%29%22%22=%22%22sqrt%284%2F9%29%22%22=%22%222%2F3

Thus the range is 0%3C=%22f%28x%29%22%2F%22g%28x%29%22%3C=2%2F3 or in interval notation: [0,2/3]

Edwin

Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!

I made corrections on the above.

Edwin

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


sqrt%284-x%5E2%29%2Fsqrt%289-x%5E2%29

Use the property sqrt%28A%29%2Fsqrt%28B%29=sqrt%28A%2FB%29 (for A and B positive):

sqrt%28%284-x%5E2%29%2F%289-x%5E2%29%29

Rewrite the radicand:

sqrt%28%28%289-x%5E2%29-5%29%2F%289-x%5E2%29%29

sqrt%281-5%2F%289-x%5E2%29%29

This expression is the square root of something less than 1.

Its minimum value is 0, when the radicand is zero:

1-5%2F%289-x%5E2%29=0
5%2F%289-x%5E2%29=1
9-x%5E2=5
x%5E2=4

The lower end of the range of the given function is 0, when x is 2 or -2.

The maximum value of the radicand is when 5%2F%289-x%5E2%29 is a minimum -- i.e., when the smallest possible value is subtracted from 1. That is when x is 0; the upper end of the range of the given function is

sqrt%281-%285%2F%289-0%29%29%29=sqrt%281-5%2F9%29=sqrt%284%2F9%29=2%2F3

ANSWER: the range of the given function is [0,2/3]

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutors have done a great job.
I know the instructions specifically state "no graphing", but let's look at a graph anyway to help verify the range is [0, 2/3] aka 0+%3C=+y+%3C=+2%2F3



The green curve is
The blue horizontal line is y = 2/3 to help show the curve maxes out at this y value.

For some reason, the built-in algebra.com graphing tool shown above is glitching a bit.
The part of the curve on the right needs to extend down to the point (2,0).
Perhaps a better look would be to use something like Desmos
https://www.desmos.com/calculator/nxfzbzullv

GeoGebra is another useful graphing tool.
There are many others.